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The concept of discounting has been developed by Magill (1970) for practical purposes by financiers and accountants, which was further introduced by Koopmans (1960) into economic theory. The usage of a discounted cost function seems to produce inconsistencies if the utility function is nonlinear, is that, if discounting is motivated by the fact that capital can grow by compound interest. Thereafter, there is an indication that one has the alternatives of operating the enterprise which is being controlled or of letting one's capital grow at a constant interest rate. Financial problems are often based on a mathematical structure with imprecise input data. Geoffard (1996) introduced the discounting and optimizing by using capital accumulation problems as variational minimax problems.
Fuzzy set theory introduced by Zadeh (1965) has been widely used to solve many practical problems, including financial risk management, engineering, business, and natural sciences since it allows us to describe and treat imprecise and uncertain elements present in a decision problem. Thereafter, the imperfect knowledge of the returns on the assets and the uncertainty involved in the behavior of financial markets may also be introduced by means of fuzzy quantities and/ or fuzzy constraints. The processing time of a job vary in many ways and may be lead to different work places. To avoid these factors, the processing time of a job are represented in the form of piecewise quadratic fuzzy numbers. A linguistic variable is a variable whose values are words or sentences in an artificial or natural language. Zadeh (1975) introduced the fuzzy concept of a linguistic variables and its application to approximate reasoning. The theory of fuzzy set has been applied in many real life applications to handle the uncertainty by several researchers like Buckley (1987), Zimmermann (1991), Choobineh & Behrens (1992), and others. Moore (1995) discussed the arithmetic of interval numbers and their applications in various domains. Abbasbandy & Amirfakhrian (2006a) studied the nearest approximation of a fuzzy quantity. They conducted an experimental study for the approximation of the parametric fuzzy numbers with the polynomial parametric fuzzy numbers. The nearest trapezoidal form of a generalized left right fuzzy number was discussed by Grzegorzewski (2002), Abbasbandy & Amirfakhrian (2006b), and later followed by Coroianu et al. (2013), and lots of others researcher working in that field. Grzegorzewski & Pasternak (2014) proposed the natural trapezoidal approximations of fuzzy numbers. They provided numerous applications in decision science, financial engineering, portfolio management, etc. Thiripurasundari el al. (2019) presented a study on arithmetic operations of type-2 triangular mixed fuzzy numbers.
Several authors applied the generalized trapezoidal fuzzy numbers to solve the decision making problems. Jain (2010) proposed the close interval approximation of piecewise quadratic fuzzy numbers for fuzzy fractional program, which was further studied by Sen et al. (2013) with applications in various domains of operations research, finance, and management. Kumar (2016) presented a conceptual model for automation of product dynamic pricing as well as the promotion in sales for a retail company. A fuzzy approach to replacement problem was developed by Biswas & Pramanik (2011). They studied the replacement problem with uncertainty, where the capital cost, scrap value or salvage value, maintenance cost or operating cost, and rate of interest were represented by imprecise values.